We consider the problem of finding a k × k submatrix of an n × n matrix with i.i.d. standard Gaussian entries, which has a large average entry. It was shown in [BDN12] using non-constructive methods that the largest average value of a k × k submatrix is 2(1 + o(1)) log n/k with high probability (w.h.p.) when k = O(log n/ log log n). In the same paper an evidence was provided that a natural greedy algorithm called Largest Average Submatrix (LAS) for a constant k should produce a matrix with average entry at most (1 + o(1)) 2 log n/k, namely approximately √ 2 smaller, though no formal proof of this fact was provided.In this paper we show that the matrix produced by the LAS algorithm is indeed (1+o(1)) 2 log n/k w.h.p. when k is constant and n grows. Then by drawing an analogy with the problem of finding cliques in random graphs, we propose a simple greedy algorithm which produces a k × k matrix with asymptotically the same average value (1 + o(1)) 2 log n/k w.h.p., for k = o(log n). Since the greedy algorithm is the best known algorithm for finding cliques in random graphs, it is tempting to believe that beating the factor √ 2 performance gap suffered by both algorithms might be very challenging. Surprisingly, we show the existence of a very simple algorithm which produces a k × k matrix with average value (1 + o k (1))(4/3) 2 log n/k for in fact k = o(n).To get an insight into the algorithmic hardness of this problem, and motivated by methods originating in the theory of spin glasses, we conduct the so-called expected overlap analysis of matrices with average value asymptotically (1 + o(1))α 2 log n/k for a fixed value α ∈ [1,√ 2]. The overlap corresponds to the number of common rows and common columns for pairs of matrices achieving this value (see the paper for details). We discover numerically an intriguing phase transition at α * 5 √ 2/(3 √ 3) ≈ 1.3608.. ∈ [4/3, √ 2]: when α < α * the space of overlaps is a continuous subset of [0, 1] 2 , whereas α = α * marks the onset of discontinuity, and as a result the model exhibits the Overlap Gap Property (OGP) when α > α * , appropriately defined. We conjecture that OGP observed for α > α * also marks the onset of the algorithmic hardness -no polynomial time algorithm exists for finding matrices with average value at least (1 + o(1))α 2 log n/k, when α > α * and k is a growing function of n.
We propose to measure intensity transmission matrices or point-spread-function (PSF) of diffusers via spatial-correlation, with no scanning or interferometric detection required. With the measured PSF, we report optical imaging based on the memory effect that allows tracking of moving objects through a scattering medium. Our technique enlarges the limited effective range of traditional imaging techniques based on the memory effect, and substitutes time-consuming iterative algorithms by a fast cross-correlation deconvolution method to greatly reduce time consumption for image reconstruction.
In some cases of imaging, wide spatial range and high spatial resolution are both required, which requests high performance of detection devices and huge resource consumption for data processing. We propose and demonstrate a multi-scale adaptive imaging method based on the idea of computational ghost imaging, which can obtain a rough outline of the whole scene with a wide range then accordingly find out the interested parts and achieve high-resolution details of those parts, by controlling the field of view and the transverse coherence width of the pseudo-thermal field illuminated on the scene with a spatial light modulator. Compared to typical ghost imaging, the resource consumption can be dramatically reduced using our scheme.
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